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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

Mathematical Swimmer

Stage: 3 Challenge Level:

Every day I go to the swimming pool and swim the same number of
lengths. I like to count the number of lengths I've done as I go as
a fraction of the total number of lengths I'm going to do that
day.

If I swam ten lengths a day, after five lengths I would say to
myself, "I've managed $\frac{5}{10}$ of my day's swimming - that's
$\frac{1}{2}$!" After eight lengths, I would say I'd done
$\frac{8}{10}$, which simplifies down to $\frac{4}{5}$. After nine
lengths, I'd say I'd done $\frac{9}{10}$, which does not
simplify.

I don't swim ten lengths a day. In fact, the total number of
lengths I swim each day is rather special.

Each number of lengths I swim will either be a prime number or the
fraction it makes of the total number of lengths will
simplify.
It is, in fact, the largest number possible for which this is
true.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.